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My supposition is; can an abstract possess an abstract? That is, a person (tangible) can possess morality or happiness, but "time" can not possess either. Or, a "society" can be said to be moral (or immoral) but is it the "society" that possesses that morality, or just the tangible members of that society?

In my opinion, the best way to think of properties (attributes, characteristics, traits) is to think of them as abstract objects. On this way of thinking of them, anything at all that possesses a property possesses (or, maybe better, instantiates) an abstract object. You possess the property of being human: you instantiate the abstract object humanity. But abstract objects themselves can also possess properties -- most obviously, the property of abstractness. On this view, society (construed as an abstract object) can be (say) immoral provided it makes sense to describe an entire society that way: any obstacle to a society's counting as immoral wouldn't stem from the abstractness of society or the abstractness of immorality.

This is not a factual question of whether conscious being can be aware of it´s own existence in the world. Rather how the chain of reasoning can be non-contradictory if one is to assume the world exists, and that this world is not a part of oneself. Consider the following:
Do I or do I not exist? I exist and there exists also something which I am not. Does the "something which I am not" exist if I do not exist?(a question as to whether the world is not me) Well if it is not a part of me, then it would surely be possible for it to exist if I do not. But if I do not exist, the world does not exist, for if the necessary perspective of observation is the perspective of the observer then the facts existing are only those which the observer can yield true or false. Therefore there can be nothing that exists when I do not exist and, stretching it further, there exists nothing which I am not.
I do not believe that www.askphilosophers.org and this computer are a product of my imagination, so please, explain how...

(1) the necessary perspective of observation is the perspective of the observer.

Therefore (2) the facts existing are only those which the observer can yield true or false.

Therefore (3) if I do not exist, the world does not exist.

If I understand correctly what you mean with these sentences, then I think there are two problems with your reasoning. The premise (1) states that observation requires an observer. Fair enough. From this you want to conclude that (2) things can exist or facts can obtain only if there is an observer who judges them to exist/obtain. But this conclusion does not really follow. Without an observer, the Rocky Mountains would not be observed or known, and the fact that there are these huge mountains would not be known to obtain. But not being known is not the same as not existing. It does not follow from the fact that mountains are not perceived by anyone that these mountains do not exist. How would the removal of all observers alter the fact that there is this mountain chain which we call the Rocky Mountains? To be sure, without observers, this mountain chain would have no name. But it could still be there, couldn't it? This is the first problem with your reasoning.

Suppose, on the contrary, that (2) any thing can exist and any fact can obtain only if observed by some observer. Even then it does not follow that this observer must be you. It could be I, for example. Well before you were born, I traveled to Colorado and carefully looked at the Rocky Mountains. According to your second proposition, the Rockies existed and various facts about them obtained while I was looking. But you didn't exist then -- and might easily never have come into existence. So it would seem that things other than you (the Rockies, the world, I) can exist independently of your existence. This is the second problem with your reasoning.

Is similarity a fact of things in the world, or is it an observation made by sentient beings?
Take two cats, for example. Is it an objective fact of the world that the two cats are similar (shape, size, biology, etc.)? Or are there, ontologically speaking, just two phenomena (or two portions of the phenomenal world) that we, as conscious beings, perceive as similar and categorize as cats?

I think it depends on what's meant by 'similarity'. If similarity is just the sharing of properties -- having in common this or that attribute -- then it would seem that any two things are similar. Even Barack Obama and the Battle of Hastings have lots of properties in common: being known to historians, being the subject of books and articles, being distinct from the number 3, being referred to by me in this sentence, and so on. (You might reply that similarity is only the sharing of intrinsic properties, but it isn't always easy to draw a line between intrinsic and extrinsic properties.)

So from the perspective of the world, any two things are similar -- and maybe equally similar, since any two things share infinitely many properties. But almost all of those properties will be uninteresting to us, and that's where we come in. From among those infinitely many properties, we conscious beings focus on just a handful in accordance with our interests. If we restrict ourselves to those interest-relative properties, then two cats will share more properties (and hence will be more similar) than a cat and a color will. But that difference seems dictated by us and not by the world independently of us.

Is modern philosophy too abstract? I mean when it asks questions about being does it ask questions that about any kind of being when perhaps it could be asking question about the particular kind of being that we live in? I guess you could say the answer is no because philosophers deal with questions about science and science is about the world we live in. But is the kind of being of science the only "concrete" form of being that philosophers can ask about? I personally think that their is more to being than either physics or hyper-abstractions that only look at being in terms of temporarily, causality and quantity, etc. Is a disagreement about what we think is "being" perhaps one of the central splits between analytic and "continental" philosophy?

I tend to use the noun 'being' as a count noun: You and I are both beings; maybe the number seven is also a being (although of a different kind from you or me). I'll therefore use the words 'existence' or 'reality' for what you seem to refer to by 'being' in your question. When it asks questions about existence or reality, modern-day philosophy -- including analytic philosophy -- ranges as broadly as you like. Philosophy doesn't confine itself to the world described by natural science. Often philosophy asks about the existence or reality of non-natural beings such as abstract objects (maybe numbers, properties, propositions) or concrete, non-natural beings (maybe immaterial minds or souls, maybe God). It's true that analytic philosophers tend to respect natural science, but they shouldn't (and largely don't) think that all legitimate questions are questions for natural science. Furthermore, contemporary philosophy -- perhaps especially analytic philosophy -- asks about ways that reality could have been but isn't: for example, in analyzing counterfactual conditionals, identity, cause and effect, the concept of knowledge, the concept of merit or desert, and countless other things too. I think contemporary analytic philosophy is much less narrowly scientistic (i.e., uncritically science-worshiping) than you may have been led to believe. For just two of many examples of analytic philosophy venturing beyond the realm of natural science, see these entries in the excellent Stanford Encyclopedia of Philosophy (an online resource I keep recommending!):

Some random suggestions: (1) David Pears pointed out that even if Kant's argument were wholly clear and wholly successful, which it is not, it could only show that existence is not an ordinary predicate, if it is a predicate. His view is that it is a predicate, just a very peculiar one; (2) There is also the view of the celebrated logician, mathematician and philosopher Bolzano, who writes in the Theory of Science ("Kinds of Propositions") that 'I take being [Sein] or actuality [Wirklichkeit] to be precisely what language makes it out to be, namely an attribute; whoever denies this confuses (I believe) actuality with substance. By substance I mean an actuality which is not an attribute of another actuality; hence I admit that we cannot truly predicate the putative abstractum of the substance (substantiality) of any object. For it is part of the concept of substance that there is no property of this kind. But it is not the same with actuality, which I consider to be a mere attribute, not only of substance itself but of each of its attributes, since every attribute of an actual thing is itself actual. And since every attribute of an object can be ascribed to it in a judgment of the form 'A has b', why not the attribute of actuality?' (3) There is a related argument deriving from Russell's Theory of Descriptions in my own Philosophical Propositions, despite the fact that Russell himself took the implication of the theory to be that the ontological argument is no good; (4) There is a defence of a stripped-down version of the ontological argument by the late Gary Matthews and Lynn Baker Rudder in Analysis for 2010.

What is the meaning of life? What is the purpose of anything existing? Does existence exist for no apparent logical and answerable reason and therefore does not need an explanation and simply is a product of random, anomalous events, or does existence exist because there is a purpose or reason for me and existence to exist? I tend to think if there is a purpose behind existence there must be something guiding existence because existence has a purpose otherwise why exist at all. Am I alive and self aware and exist because something made me exist or am I the result of a randomness of phenomenon that allowed me to develop the conscious ability to question my existence and therefore find some justification for my existence even though the questioning of existence is pointless in any case? In other words do I and everyone else exist for a reason or is there meant to be no apparent reason for my existence therefore I am allowed free reign to believe I exist for some apparent reason which may or may not be a...

You certainly have asked THE big question! Many religious thinkers do believe that there is a meaning to life and a purpose as well. For a good representation of a broadly Christian point of view (but one that would be satisfying to traditional Jews, Muslims, and some Hindus) you might check out Mark Wynn's book God and Goodness. In this philosophy, you and the cosmos as a whole exist because it is good that you and the cosmos exist; moreover, it is created by an all good God whose purpose for creating was to being about goodness. I personally adopt such a position, but many fellow philosophers do not, either because they simply deny that there is a God or they are suspicious about objective values like goodness. But leaving aside religious concerns, if you simply recognize values like happiness (or flourishing) then you will find yourself among many philosophers (religious and secular) who think that a big part or the meaning of life (its point) is for there to be human flourishing, and going beyond that a flourishing of the whole community or body of living things that make up our planet. Aristotle is a good source on that, and a modern defender of happiness as the meaning of life today is Stewart Goetz.

Some of the things you might want to distinguish in your question: when someone asks for the meaning of life, I suggest two questions are at issue: the person is asking "what exists?" and "what should I value?" The first question is (I think) unavoidable and it has an answer even if no one knows that that answer is. In other words, either there is some kind of God or not, either there is some purpose for the cosmos or not, and so on... The second may lead in two directions. The first is the one I take which is that there really are goods and ills (justice and friendship really should be valued and cruelty and injustice should really be avoided or fought). A second approach is more skeptical and assumes that there are no real objective values. On this account, values might actually simply come down to felt, changing preferences. Then there is also a middle of the road: some things are objectively good and somethings (like whether you choose to have a romantic partner or a lifetime of celebacy --as the great scientist Newton chose and was proud of it).

Good wishes! For another great book on all this, check out Thomas Nagel's short book on what it all means (approximate title).

Why not? Both vegetables and us having being, albeit rather different.

I recall a show many years ago which had a very human plant in it, but the trouble with it for our purposes here is that really it was a human being dressed up as a plant, as far as I could see from its behaviour, and so hardly expressive of a very different life form.

Is it a logical contradiction for something to come from nothing? I've heard that this causal principle is intuitive and something a rational person cannot deny. However, is it metaphysically possible for something to indeed come from nothing? Is that a logical contradiction concerning cause and effect? If we're not strictly talking about cause and effect, is it still possible for something to come from nothing? Is an event always contingent upon a cause?

Great question. I don't have an answer. But some thoughts depend on how you frame things. If by "causation" you have a certain model in mind (e.g. where something is transferred from cause to effect) then it does seem contradictory to say that 'something comes from nothing' -- if that is taken to mean 'nothing causes something to come into being' -- for that seems to require both that something be transferred from cause to effect (by the word 'causes') yet that there be nothing to be transferred (since 'nothing' is said to be THE cause) .... But who is to say that 'causation' should be understood on that model? And even that model would not rule out the metaphysical possibility of something coming from nothing, if what that means is 'something comes into existence uncaused' -- there does not seem to be a contradiction, or at least not an obvious one, in the latter, since no 'causation' is being implied .....

Suppose I tell my friend that leprechauns don't exist. He responds: "Well, not in THIS realm, they don't. But they MIGHT exist in some hitherto undiscovered realm."
To what extent does the claim 'X exists' depend on its being discoverable, or knowable? As a curious person, this question has really bothered me the past few days. There's something comforting about having knowledge, and that there might be an infinite amount of unknowables is rather disconcerting to me. Does Ayer's position -- that for a claim to be meaningful it must either be tautological or empirically veriable -- apply here?
If someone could shed some light on this quandary, I'd be immensely appreciative. I really don't know my I allow myself to be bothered my these types of philosophical questions.

While Ayer's verificationism has gone out of fashion (he and others could not settle on a formulation of it that did not rule out science or some such apparently meaningful discourse) there are forms of what is called anti-realism which define 'truth' in terms of warranted assertability, which would rule out the possibility of there being truths that are out of reach from what we can know (at least in principle). Alas, there is a good argument against such a position in Thomas Nagel's work The View From Nowhere.

One other idea to consider is that your friend may be right but in a way that has nothing to do with THIS (our) world. Some philosophers (David Lewis etc) have argued that there are indefinitely many POSSIBLE WORLDS. So, you might reply that, yes, leprechauns actually do exist but in a possible world not remotely related to ours! Check out Lewis's book on the plurality of worlds. It is awesome.